The signature(p, q, r) of a metric tensorg (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrixgab of the metric tensor with respect to a basis. Alternatively, it can be defined as the dimensions of a maximal positive, negative and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (p, q) implying r = 0 or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signature (1, 3) resp. (3, 1).[1]

There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as p − q, where p and q are as above, which is equivalent to the above definition when the dimension n = p + q is given or implicit. For example, s = 1 − 3 = −2 for (+, −, −, −) and s = 3 − 1 = +2 for (−, +, +, +).

Given a finite-dimensional real vector spaceV with a metric tensor (or scalar product) g, then for every orthogonal basis of V, the metric applied to each basis vector eμ, i.e. g(eμ, eμ), 1 ≤ μ ≤ n will produce a value that is a positive, negative or zero. By Sylvester's law of inertia, the number of values of each of these three cases is independent of the choice of orthogonal basis. The signature (p, q, r) of g is the number of positive, negative and zero values respectively. When r is nonzero, the metric tensor g is called degenerate; when q = r = 0, g is called positive definite; and when p = r = 0 it is called negative definite.

Sylvester's law of inertia: independence of basis choice and existence of orthonormal basis[edit]

According to Sylvester's law of inertia, the signature of the scalar product (a.k.a. real symmetric bilinear form), g does not depend on the choice of basis. Moreover, for every metric g of signature (p, q, r) there exists a basis such that gab = +1 for a = b = 1, ..., p, gab = −1 for a = b = p + 1, ..., p + q and gab = 0 otherwise. It follows that there exists an isometry(V1, g1) → (V2, g2) if and only if the signatures of g1 and g2 are equal. Likewise the signature is equal for two congruent matrices and classifies a matrix up to congruency. Equivalently, the signature is constant on the orbits of the general linear group GL(V) on the space of symmetric rank 2 contravariant tensors S2V∗ and classifies each orbit.

The number p (resp. q) is the maximal dimension of a vector subspace on which the scalar product g is positive-definite (resp. negative-definite), and r is the dimension of the radical of the scalar product g or the null subspace of symmetric matrixgab of the scalar product. Thus a nondegenerate scalar product has signature (p, q, 0), with p + q = n. The special cases (n, 0, 0) and (0, n, 0) correspond to positive-definite and negative-definite scalar products which can be transformed into each other by negation.

For a symmetric matrix, the characteristic polynomial will have all real roots whose signs may in some cases be completely determined by Descartes' rule of signs.

Lagrange algorithm gives a way to compute an orthogonal basis, and thus compute a diagonal matrix congruent (thus, with the same signature) to the other one: the signature of a diagonal matrix is the number of positive, negative and zero elements on its diagonal.

According to Jacobi's criterion, a symmetric matrix is positive-definite if and only if all the determinants of its main minors are positive.

the metric signature is (1, 3, 0), since it is positive definite in the time direction, and negative definite in the three spatial directions x, y and z. (Sometimes the opposite sign convention is used, but with the one given here s directly measures proper time.)

If a metric is regular everywhere then the signature of the metric is constant. However if one allows for metrics that are degenerate or discontinuous on some hypersurfaces, then signature of the metric may change at these surfaces.[2] Such signature changing metrics may possibly have applications in cosmology and quantum gravity.